Sine: carrying the opposite across
On the unit circle the hypotenuse is 1, so sinθ = opposite ÷ hypotenuse = opposite. The vertical red side is the sine. Drag the point round, or tap Step +15°, to carry that height straight across onto the graph.
Why is the opposite equal to sinθ?
In any right-angled triangle SOH tells us sinθ = opposite / hypotenuse. On the unit circle the radius (the hypotenuse) is set to 1, so dividing by 1 changes nothing, and the opposite side and sinθ are the same length.
As the radius sweeps round, the height of the point above the horizontal axis is exactly sinθ. Plotting that height against the angle traces the sine wave. Past 360° the radius is back where it started, which is why the graph repeats every 360°.
Cosine: the adjacent side
Cosine is the horizontal side: cosθ = adjacent ÷ hypotenuse = adjacent. The blue length is read off and stood up vertically just outside the circle, then carried across. Notice the cosine graph is just the sine graph translated: cosθ ≡ sin(θ + 90°).
Why is the cosine graph a shifted sine graph?
Cosine measures the horizontal distance of the point from the centre. As the radius turns, that horizontal distance changes in exactly the same way as the height does, but a quarter turn earlier.
Because the adjacent side leads the opposite side by a quarter turn, the whole cosine curve sits 90° ahead of the sine curve.
Tangent: the line that touches the circle
Draw the vertical line that just touches the circle at (1, 0), the tangent line. Extend the radius until it hits that line: the radius extended is the secant and the piece of the tangent line it cuts off is tanθ. As θ approaches 90° the radius becomes parallel to the tangent line, so the two never meet and tanθ increases without bound.
Sketching a reciprocal graph
Explanation
Solutions from graphs
Type an angle for the primary value P. The horizontal line shows every angle with the same value: the secondary value S comes from the symmetry of the curve, and the rest are R, found when the graph repeats.
All solutions of the equation
In the range −360° to 720° (to 2 d.p.):
sin and cos between 0° and 90°
Between 0° and 90° the sine and cosine curves are mirror images in the line θ = 45°. Reflecting there swaps them, which is the cofunction rule.
CAST diagram
The unit circle is the graph wrapped round a circle. In each quadrant one set of functions is positive: All, Sin, Tan, Cos. Drag the point: the primary P and its secondary S have the same value, and the height (or width) carries straight across to the graph.
Finding the point on the graph
Secondary and repeated values
Inverse trig graphs
An inverse function undoes its parent, and its graph is the parent reflected in the line y = x. But sin, cos and tan repeat, so they are not one-to-one. We first restrict each to a stretch where it is one-to-one (the bold part); only that part has an inverse. The two notations and mean exactly the same thing.
arcsin x, sin-1x and (sin x)-1
Why restrict the domain?
Switching the axes (how to sketch it)
The quickest way to sketch an inverse graph: first sketch the restricted parent graph, then swap the axes (reflect in y = x) and fill in the same shape. Every key point just has its coordinates switched, for example on sine becomes on arcsine.
Does the composition undo to x?
Exact values
Two reference triangles give you every exact value. Split an equilateral triangle of side 2 down the middle for 30° and 60°, and use a right-angled isosceles triangle with sides 1, 1, √2 for 45°. Hover over any value in the table to see where it comes from (0° and 90° are read off the graph instead).